Null Sectional Curvature Pinching for Cr-lightlike Submanifolds of Semi-riemannian Manifolds

In this article we obtain the pinching of the null sectional curvature of CRlightlike submanifolds of an inde…nite Hermitian manifold. As a result of this inequality we conclude some non-existence results of such lightlike submanifolds.Moreover using the Index form we prove more non-existence results for CR-lightlike submanifolds. 1. Introduction The theory of submanifolds of a Riemannian or semi-Riemannian manifold is well-known .(see for example, [1] and [6]). However the geometry of lightlike (null) submanifolds (for which the geometry is di¤erent from the non-degenerate case) is highly interesting and in a developing stage. In particular, curvature pinching relations are of substantial interest as they give the bounds for the curvature.Analogous to sectional curvature in Riemannian case, Duggal [2] de…ned the null sectional curvature for lightlike submanifolds. Earlier on, A. Gray [4] investigated di¤erent pinchings for sectional and bisectional curvature under certain conditions in case of Kaehler manifolds. In this article we would like to study CR-lightlike submanifold of an inde…nite almost Hermitian manifold (for de…nite Hermitian manifolds see [5]) and hence obtain the null sectional curvature pinching K (JY ) K (X) 3K (JY ) as our main theorem; where X;Y are two orthonormal vectors in some distribution of S(TM) and K (X) is the null sectional curvature [2] . With the help of this null curvature pinching we obtain some non-existence results for CR-lightlike submanifolds of an inde…nite almost Hermitian manifold. In the last we also study some applications of the Index form and Jacobi equation [6], to conclude some more non-existence results. 2. Preliminary Let ( M; g) be an (m + n) dimensional semi-Riemannian manifold and g be a semi-Riemannian metric on M . Let M be a lightlike submanifold of M: 2000 Mathematics Subject Classi…cation. 53C50, 53B30 . Key words and phrases. CR-Lightlike submanifolds, Null sectional curvature, Index form, Variation of a curve. Submitted April 25, 2012. Published October 8, 2012. This Research Work is supported by UGC Major Research Project No: 33/112(2007)-SR. 108 NULL SECTIONAL CURVATURE PINCHING 109 De…nition 2.1. [2] A lightlike submanifold M of an inde…nite almost Hermitian manifold M is said to be CR-lightlike submanifold if and only if the following two conditions are ful…lled: (a) J(RadTM) is a distribution on M such that RadTM \ J(RadTM) = f0g; (b) there exists vector bundles S(TM); S(TM?); ltr(TM); D and D0 over M such that S(TM) = fJ(RadTM) D0g ? D ; JD = D ; J(D0) = L1 ? L2 where D is a non-degenerate distribution on M , and L1 and L2 are vector subbundles of ltr(TM) and S(TM?) respectively. It is seen that there exists examples of CR-lightlike submanifolds of an inde…nite Hermitian manifold. An example of such kind can be given as follows[2]: Example1. Let M be a submanifold of codimension 2 of R 6 given by the equations x = x cos x sin f(x; x) tan ; x = x sin + x cos + f(x; x) where 2 R 2 + k ; k 2 Z and f is a smooth function such that @f @x3 ; @f @x4 6= (0; 0). It is easily veri…ed that the tangent bundle of M is spanned by f = @ @x1 + cos @ @x5 + sin @ @x6 ; X = @ @x2 sin @ @x5 + cos @ @x6 ; X1 = @ @x3 @f @x3 tan @ @x5 + @f @x3 @ @x6 ; X2 = @ @x4 @f @x4 tan @ @x5 + @f @x4 @ @x6 g: Then M is a 1-lightlike submanifold with Rad(TM) = spanf g. Moreover by using J(x1; y1; :::::; xm; ym) = ( y1; x1; :::::; ym; xm) we obtain that JRad(TM) is spanned by X and therefore it is a distribution on M . Hence M is a CR-lightlike submanifold of codimension 2 of R 2. Example 2. We consider a submanifoldM of codimension 2 in R 2 given by the equations x = x cos x sin xx tan ; x = x sin + x cos + xx where 2 R 2 + k ; k 2 Z . Then TM is spanned by U1 = (1; 0; 0; 0; 0; 0; cos ; sin );U2 = (0; 1; 0; 0; 0; 0; sin ; cos ); U3 = (0; 0; 1; 0; 0; 0; 0; 0);U4 = (0; 0; 0; 1; 0; 0; 0; 0); U5 = (0; 0; 0; 0; 1; 0; x tan ; x);U6 = (0; 0; 0; 0; 0; 1; x tan ; x): It is easy to check that this submanifold is 1-lightlike submanifold of R 2 such that Rad(TM) = spanfU1g: Furthermore by using J(x1; y1; :::::; xm; ym) = ( y1; x1; :::::; ym; xm) 110 MOHAMMED JAMALI AND MOHAMMAD HASAN SHAHID on R 2 we see that U2 = JU1. This shows that JRad(TM) is a distribution on M . Hence M is a CR-lightlike submanifold. Let u 2 M and be a null vector in TuM . A plane P of TuM is called a null plane directed by if it contains , gu( ;X) = 0 for any X 2 P and there exists X 2 P such that gu(X ; X ) 6= 0: As in case of lightlike submanifolds the collection of null vectors is denoted by Rad(TM) and non-null vectors by S(TM) i.e. we always have 2 Rad(TM) and X 2 S(TM): This means that in case of lightlike submanifolds null plane is spanned by a vector of Rad(TM) and a vector of S(TM): De…nition 2.2. [2] The null sectional curvature of P with respect to and r is de…ned as the real number K (X) = g(R(X; ) ;X) g(X;X) ;8 2 Rad(TM); X 2 S(TM): The null sectional curvature of P with respect to and r is de…ned as the real number K (X) = g( R(X; ) ;X) g(X;X) ;8 2 Rad(TM); X 2 S(TM): We denote by Q (X); the quantity g(R(X; ) ;X) i.e. Q (X) = g(R(X; ) ;X) which gives Q (X) = kXkK (X): (2.1) Similarly we have Q (X) in case of M: In [3] Duggal and Jin de…ned totally umbilical lightlike submanifolds of a semiRiemannian manifold. De…nition 2.3. [2] A lightlike submanifold M of a semi-Riemannian manifold M is totally umbilical if there is a smooth transversal vector …eld H 2 (tr(TM)) onM called the transversal curvature vector …eld of M , such that for all X;Y 2 (TM) ; h(X;Y ) = Hg(X;Y ): A CR-lightlike submanifold which is totally umbilical is called totally umbilical CR-lightlike submanifold. The following lemma is an important result regarding the null sectional curvature of totally umbilical CR-lightlike submanifold: Lemma 2.4. Let (M; g) be a totally umbilical CR-lightlike submanifold of an almost Hermitian manifold ( M; g). Then the null sectional curvature of M is equal to the null sectional curvature of M: Proof. Let (M; g) be a totally umbilical CR-lightlike submanifold of ( M; g): Then from [2] we can write g( R(X; ) ;X) = g(R(X; ) ;X) + g(h(X; ); h( ;X)) g(h(X;X); h( ; )); (2.2) 8 X 2 D and 2 Rad(TM):Since M is totally umbilical we have,